Supply schedule for an individual: what precisely do the data points mean?

Working through ‘Lessons for the Young economist’ at the moment, but having some difficulty with the concept of supply schedules for individuals. It seems to me that the definitions i’ve seen (both in LFTYE and online) are ambiguous, and I’d like to clear that up.

In LFTYE Supply is definined as follows:

The relationship between the price of a good (or service), and the number of units that producers want to sell at each hypothetical price.

This makes sense to me when simultaneously considering many producers who become willing to start selling ther products at different prices (given the different input costs, and correspondingly different prices at which profit can be made), giving a rising supply curve. But I don’t understand exactly what a supply schedule means for an individual.

Specifically my confusion has to do (i think) with the phrase ‘number of units a producer wants to sell’.

As soon as the price is higher than the production cost, wouldn’t an individual producer ‘want to sell’ an infinite amount of units? If this was the case though, a supply curve for an individual would always look like a single step, rather than a slope. I have the impression this isn’t right, and that a typical supply curve for an individual is also likely to more closely resemble a slope (is that correct?).

So I’m suspecting that ‘the number of units a producer wants to sell at a range of prices’ can be more precisely put as something like ‘the number of units a producer would be willing and able to sell at a range of prices, at a given moment’. So his inventory of the supplier at that given moment is an important constraint: He’d be willing and able to sell fewer units at a low price and more units at a high price to avoid shortages and maximise revenue. Have I got that right?

If this is the case, then is it also true that a generic supply curve could be conceptualised (although not drawn) as end with an upward tick, a line that extends upwards forever, representing the limit of the suppliers inventory, beyond which he’s not able to sell more units at any price?

Can anyone help me out here with confirmations/corrections here? My thanks in advance.